We discuss and prove validity of the Hubbard-Stratonovich (HS) identities over hyperbolic domains which are used frequently in the studies on disordered systems and random matrices. We also introduce a counterpart of the HS identity arising in disordered systems with "chiral" symmetry. Apart from this we outline a way of deriving the nonlinear $\sigma$-model from the gauge-invariant Wegner $k-$orbital model avoiding the use of the HS transformations.
Categories: math-ph cond-mat.mes-hall hep-th math.MP